Unpolarisierte Neutronen- Van Hove Streufunktion S(κ,ω) Aufteilung von S: elastische – inelastische Streufunktion

A short Excursion to Fourier And Delta Functions .... it follows by extending the range of x to more than –L/2 ...L/2 and going to 3 dimensions (v0 the unit cell volume)

Neutronen – Diffraktion Gitter G mit Basis B: Gitterfaktor Strukturfaktor „Isotopen-inkoherente-Streuung“ „Spin-inkoherente-Streuung“ unabhängig von κ: ein Element(NB=1):

Gitterfaktor Strukturfaktor The Nobel Prize in Physics 1994 "In simple terms, Clifford G. Shull has helped answer the question of where atoms are, ...“, (Nobel citation)

Bragg’s Law in Reciprocal Space (Ewald Sphere) 2q Scattered Neutron τ q Incoming Neutron

Einkristall- diffraktometrie E2 – HMI, Berlin τ O

Bragg’s Law in Reciprocal Space (Ewald Sphere) sin OB τ = sin q = O τ /OB = O τ /(2p/l) O sin q = (O τ/2p)l Reflecting Plane 2p/l q q But since τ is a reciprocal lattice point, the length O τ is by definition equal to 2p/dhkl C 2q q τ sin q = (1/2dhkl) l B Neutron 2dhkl sin q= l sin q / l = (1/2dhkl) = (1/2)(1/dhkl) = (1/2)shkl

Pulver- diffraktometrie D1B guide hall n°1, thermal guide H22 monochromator take-off angle 44.22° crystal pyrolytic graphite (002) . wavelength 2.52 Å . flux at sample/n cm-2s-1 6.5 x 106 Germanium (311) 1.28 Å 0.4 x 106 max beam size 5 x 2 cm2 angular range 2     -20° ... 144° detector 3He multidetector containing 400 cells 80° radius of curvature 1.525 m detector efficiency 60 % at   = 2.52 Å max diameter / mm available around the sample 600 sample environment cryostat 1.7 ... 300 K furnace < 800 °C < 2500 °C by special arr electromagnet 1 T; 22 mm vertical or horizontal gap

Pulver-diffraktometrie 7C2, LLB Saclay GdCu2In Pulver-diffraktometrie I(κ) [counts] 0 1 2 3 4 5 6 |κ|[Å-1] 2θ.... Streuwinkel Detektor SF.. Strukturfaktor L ... Lorentzfaktor (betont kleine Winkel) Einkristall: 1/sin2θ Pulver Zyl.:1/(sin2θ.sinθ) T Transmissionskoeffizient γ Korrektur für Extinktion

GdCu2In Gitterfaktor Strukturfaktor #lambda= 0.58 A #thetamax=18 #nat=4 nonmagnetic atoms in primitive crystallographic unit cell: #[atom number] x[a] y[b] z[c] dr1[r1] dr2[r2] dr3[r3] [Gd] 0 0 0 0 0 0 [Cu] 0.25 0.25 0.25 0.25 0.25 0.25 [Cu] 0.25 0.25 0.75 0.75 0.75 -0.25 [In] 0.5 0.5 0.5 0.5 0.5 0.5 # a=6.62 b=6.62 c=6.62 alpha= 90 beta= 90 gamma= 90 # r1x= 0 r2x= 0.5 r3x= 0.5 # r1y= 0.5 r2y= 0 r3y= 0.5 primitive lattice vectors [a][b][c] # r1z= 0.5 r2z= 0.5 r3z= 0 # nofatoms=1 number of atoms in primitive unit cell GdCu2In Gitterfaktor Strukturfaktor   h k l d[A] |kappa|[A^-1]2theta Ikern imag itot |sf| lpg } 1.000 1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 -1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 -1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 -1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 1.000 1.000 -1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 -1.000 -1.000 1.000 3.82206 1.64389 8.703 11.661 0.000 11.661 24.966 87.101 0.000 2.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 -2.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 0.000 2.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 0.000 -2.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 2.000 0.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 -2.000 0.000 0.000 3.31000 1.89820 10.053 0.025 0.000 0.025 1.336 65.389 0.000 2.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 -2.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 -2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 0.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 0.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 0.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 0.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 -2.000 2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 2.000 -2.000 0.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 -2.000 2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822 0.000 2.000 -2.000 2.34052 2.68446 14.235 83.558 0.000 83.558 109.804 32.822

Beispiel 2 In einem elastischen Streuexperiment beträgt die Einfallsenergie 63 meV. Die Gitterkonstante der kubischen Probe beträgt 0.314 nm. Kann der (430) Reflex in diesem Streuexperiment vermessen werden ?

Neutron – Photon Streuquerschnitte Vorteile von Neutronen: Kontrast bei benachbarten Elementen – man sieht z.B. Überstrukturen Leichte Elemente besser nachweisbar Isotope können unterschieden werden

Laue Methode Einkristalle „weißer“ Strahl Film oder Flächendetektor hinter der Probe schnelles Erkennen der Symmetrie - wird zum Orientieren von Einkristallen benutzt VIVALDI very-intense vertical-axis Laue diffractometer

4-Kreismethode ω φ χ Einkristalle monochromatischer Strahl ein Detektor EK in beliebige Richtungen orientierbar (Eulerwiege) ω D10 ILL

Flugzeitmethode Det Spallationsquelle (gepulst) 2θ Probe Streuwinkel fest (Vorteil z.B. bei Druckzellen) |k| wird variiert (kein Monochromator) über die Zeit (zuerst kommen die raschen, dann die langsameren Neutronen) bessere Nutzung der Quelle (keine Monochromator-verluste) Auflösung umso besser, je größer Abstand zur Quelle (HRPD: 90m)

Time-of-flight Bragg equation - 2dhklsin =  Two basic equations: where m,v = mass, velocity of neutron L = length of flight path t = time of flight of neutron

Time-of-flight equation Combine: L is a constant for the detector, h, m are constants so: t d d-spacings are discriminated by the time of arrival of the neutrons at the detector

The biggest error in the experiment is where the neutrons originate This gives an error in the flight path, L typical value ~5cm Hence as L increases, error in d is reduced - resolution of the instrument is improved e.g. instrument at 10m compared to instrument at 100m 100m = HRPD, currently highest resolution in the world

HRPD, GEM GEM General purpose Materials Diffraktometer Sample area collimators and detectors on HRPD.  Neutrons enter via the yellow flight tube on the left. GEM General purpose Materials Diffraktometer

p-dichlorobenzene (DCB) refined structure

. . A C dhkl f C = a h b k - shkl = h a * + k b * + l c * a b c/l b/k A C dhkl f C = a h b k - shkl = h a * + k b * + l c * a b C . shkl= ( - ) . (h a * + k b * + l c *)= h k b h k a - . (h a * + k b * + l c *) c *) = h + 0 + 0 – (0 + k + 0) = 1 – 1 = 0 Therefore shkl is perpendicular to C . In the same way one can show that it is perpendicular to A, therefore perpendicular to the plane

. . n = ha* + kb* + lc* |shkl| n = ha* + kb* + lc* n = |shkl| dhkl = cos f = . n = ha* + kb* + lc* |shkl| . = 2p